Page 31 - Textos de Matemática Vol. 47

and, in particular,
X
l=1 k= 1 t2T j2It ( 1)|S|+1
X1 X1 _d 1+
R(X, )=
;6=S({1,...,d} 1+
X1 X1 _ l=1 k= 1 j2S
1 ↵l,k,j j
l,k,j
R(X, )=
0
1✓ 0 d 1✓
MAX-MIN DEPENDENCE COEFFICIENTS
21
l,k,j
.
X ( 1)|T |+1 R ( X , , I ) = X1 X1 _ _
;6=T ({1,...,p} 1+ 1 ↵ t
1 + ( 1)p X1 X1 _p _
X ( 1)|T |+1 1 + ( 1)p
;=6 T({1,...,p} 1+ 1/✓|I | 1+ 1/✓|I | tt jt
l,k,j
1+
l=1 k= 1 t=1j2It 1 + ( 1)d
1 ↵ j
1 ↵ t
l=1 k= 1 j=1 Example3.2. FortheSymmetricLogisticmodel`(x1,...,xd)=⇣Pd x1/✓⌘✓
we have
R(X, ,I)= X !✓ Xp !✓,
j=1 j
t2T t=1 X ( 1)|S|+1 1 + ( 1)d
;6=S({1,...,d} 1+@X 1/✓A 1+@X 1/✓A jj
j2S j=1 Xd 1 1 1
and
R(X,1)= ( 1)k+1 dk 1+k✓ (1+( 1)d)1+d✓.
k=1
Several parametric and non-parametric estimators for the tail dependence function are available in the literature (Beirlant et al., 2004; Schmidt and Stadtmu¨ller, 2006; Krajina, 2010) which can be applied to the terms in (2.6). The comparison of estimation procedures is out of our purposes in this paper and we simply remark that the definition of the max-min dependence coe cient suggests a non-parametric estimator based on sample means.
Let X(k) = (X(k),...,X(k)), k = 1,...,n, be a sequence of independent 1d
copies of X and Fˆj the empirical distribution provided by X(k), k = 1, . . . , n,
j = 1,...,p.
j